Sorry for the undeveloped question---it seems it should be elementary, but I just can't find anything useful! A nudge in the direction of relevant theorems would be greatly appreciated.

Really the title says it all. I have $p(x) \neq q(x)$, both of degree $n$ and convex on $\mathbb{R}$ with $p(0) = q(0) = 0$. Is there any better lower bound on the number of intersections than just $n$?